Abstract: I will present a pedestrian way of introducing the concepts of Riemannian geometry for the noncommutative 4-sphere. This is done in analogy with the classical view of the 4-sphere as being embedded in five dimensional Euclidean space. By closely mimicking the construction of the tangent space for an embedded manifold in classical geometry, a particular module over the noncommutative 4-sphere is found and compared with the tangent bundle. Together with a set of corresponding derivations, one may introduce a connection on this module and show that it shares several properties with the Levi-Civita connection on the classical tangent bundle.

Abstract: In this session we will talk about rationality of algebraic tori. We will first define the notion of rational algebraic variety and then some relaxed notions of rationality. Algebraic tori are important objects in studying algebraic groups. In fact the role which they play is similar to the role of tori in the theory of Lie groups. In order to talk about the results about rationality of algebraic tori we will take a look at duality between the category of algebraic tori and category of G-lattices. We will end the session with the main results about birational classification of tori in small dimensions (up to 5).

Abstract: The $E_2$ term of the Adams spectral sequence is given by Ext groups over the Steenrod algebra, namely the algebra of primary (stable) cohomology operations. In this talk, we will present work of Baues and Jibladze on secondary chain complexes and secondary derived functors, which generalize the usual chain complexes and derived functors in homological algebra. With this machinery, the $E_3$ term can be expressed as a secondary Ext group over the algebra of secondary cohomology operations.

Abstract: Index theory on noncommutative algebras that arise from far more complicated spaces than manifolds, such as the space of leaves of a foliation, and properties of a noncommutative Chern character led to the discovery of cyclic cohomology and Connes' index formula. It states the coincidence between an analytic and a topological index for noncommutative algebras. The local index formula of Connes and Moscovici is an effective tool for computing the index pairings in noncommutative geometry by local formulas. This talk will be a review of these ideas and an indication of my joint works with Masoud Khalkhali on the computation of local geometric invariants of noncommutative tori, such as scalar curvature, when their flat geometry is conformally perturbed by a Weyl factor.